Minimization and parameter estimation for seminorm regularization models with I-divergence constraints

In this papers we analyze the minimization of seminorms ‖L · ‖ on Rn under the con-straint of a bounded I-divergence D(b,H ·) for rather general linear operators H and L. The I-divergence is also known as Kullback-Leibler divergence and appears in many models in imaging science, in particular when dealing with Poisson data but also in the case of multiplicative Gamma noise. Often H represents, e.g., a linear blur operator and L is some discrete derivative or frame analysis operator. A central part of this pa-per consists in proving relations between the parameters of I-divergence constrained and penalized problems. To solve the I-divergence constrained problem we consider various first-order primal-dual algorithms which reduce the problem to the solution of certain proximal minimization problems in each iteration step. One of these proximation prob-lems is an I-divergence constrained least squares problem which can be solved based on Morozov’s discrepancy principle by a Newton method. We prove that these algorithms produce not only a sequence of vectors which converges to a minimizer of the constrained problem but also a sequence of parameters which convergences to a regularization param-eter so that the corresponding penalized problem has the same solution. Furthermore, we derive a rule for automatically setting the constraint parameter for data corrupted by multiplicative Gamma noise. The performance of the various algorithms is finally demon-strated for different image restoration tasks both for images corrupted by Poisson noise and multiplicative Gamma noise.